A305141 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^3 dx ) / A(x), for n > 0.

1, 1, 11, 228, 6621, 240689, 10351550, 509604000, 28110904439, 1711981045939, 113863658640249, 8201890764752000, 635637023178406472, 52712939749766528868, 4656568244615480818794, 436486181882215383918344, 43268184144892865821692559, 4522468113281674174052795751, 497107356171097228291772997005

Offset: 0

Author

Paul D. Hanna, May 31 2018

Keywords

• nonn

Comments

It is remarkable that this sequence should consist entirely of integers.

Note: 0 = [x^n] exp( n * Integral F(x)^3 dx ) / F(x) holds for n > 0 when F(x) = 1 + x*F(x)^4 is a g.f. of A002293.

Examples

			O.g.f.: A(x) = 1 + x + 11*x^2 + 228*x^3 + 6621*x^4 + 240689*x^5 + 10351550*x^6 + 509604000*x^7 + 28110904439*x^8 + 1711981045939*x^9 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in exp(n^2*Integral A(x)^3 dx)/A(x) begins:
n=0: [1, -1, -10, -207, -6076, -223435, -9707184, ...];
n=1: [1, 0, -9, -616/3, -6115, -1128624/5, -88359418/9, ...];
n=2: [1, 3, 0, -535/3, -6107, -1156806/5, -455986832/45, ...];
n=3: [1, 8, 35, 0, -5257, -1167296/5, -52842348/5, ...];
n=4: [1, 15, 126, 2219/3, 0, -1003419/5, -96971176/9, ...];
n=5: [1, 24, 315, 9104/3, 22299, 0, -83502496/9, ...];
n=6: [1, 35, 656, 8883, 98045, 4304146/5, 0, ...];
n=7: [1, 48, 1215, 65480/3, 316393, 19736784/5, 1805083618/45, 0, ...]; ...
in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2*Integral A(x)^3 dx)/A(x), for n > 0.
RELATED SERIES.
A(x)^2 = 1 + 2*x + 23*x^2 + 478*x^3 + 13819*x^4 + 499636*x^5 + 21382124*x^6 + 1048225434*x^7 + 57622342803*x^8 + 3499302699294*x^9 + ...
A(x)^3 = 1 + 3*x + 36*x^2 + 751*x^3 + 21627*x^4 + 777888*x^5 + 33127964*x^6 + 1617262071*x^7 + 88594431639*x^8 + 5364836605107*x^9 + ...
exp( Integral A(x)^3 dx) = 1 + x + 7*x^2/2! + 235*x^3/3! + 19033*x^4/4! + 2701081*x^5/5! + 578096911*x^6/6! + 171419630467*x^7/7! + 66700397369425*x^8/8! + ...
A'(x)/A(x) = 1 + 21*x + 652*x^2 + 25373*x^3 + 1159491*x^4 + 60142320*x^5 + 3468823324*x^6 + 219440572309*x^7 + 15077173544671*x^8 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A305137, A305138, A305139, A305140, A305142, A305143.

Programs

PARI

{a(n) = my(A=[1],m); for(i=1,n+1, m=#A; A=concat(A,0); A[m+1] = Vec( exp(m^2*intformal(Ser(A)^3)) / Ser(A) )[m+1] );A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 2.1981... - Vaclav Kotesovec, Oct 19 2020